1. High-Quality Simplification with Generalized Pair Contractions Pavel Borodin,* Stefan Gumhold, # Michael Guthe,* Reinhard Klein* *University of Bonn, Germany # University of Tuebingen, Germany

2. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 2 Outline  Introduction  Related Work  Generalized Pair Contractions  Spatial Search Data Structure  Simplification Algorithm  Applications and Results  Conclusion

3. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 3 Introduction  Polygon models are widely used  Permanent growth of model complexity  Reduction is often necessary  Many mesh simplification algorithms Original model (34834 vertices) Simplified model (3483 vertices) More simplification (1000 vertices)

4. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 4 Introduction  Many models don't have consistent connectivity  T-vertices  self- intersections  gaps and small holes  close, but not connected surface parts

5. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 5 Introduction  Problem: vertex pair contractions not always sew together geometrically close but not incident surface parts  Unnecessarily large complexity of simplified models / unnecessarily large errors  Need of more general operations ?

6. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 6 Introduction  Problem: quadric error metric does not preserve sharp features Simplified 200 vertices Original 1972 vertices Qf1Qf1Qf1Qf1 Qf2Qf2Qf2Qf2 e Q e = Q f 1 + Q f 2

7. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 7  Edge collapse simplification [Hoppe 1996]  contracts two adjacent vertices  performs no topology alteration Related Work v new v2v2v2v2 v1v1v1v1

8. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 8  Vertex pair contraction simplification [Popovic and Hoppe / Garland and Heckbert 1997]  the contracted vertices do not necessarily lie on a common edge  allows topology modifications  sews together unconnected parts and closes small gaps Related Work v new v1v1v1v1 v2v2v2v2

9. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 9  Vertex-edge contractions on boundaries [Borodin et al. 2002 - Progressive Gap Closing]  generalizes vertex pair contraction  contracts boundary vertex with boundary edge  improves the sewing potential of vertex pair contraction Related Work v new v v int v e

10. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 10 Generalized Pair Contractions  Three new operations:  vertex-edge contraction  vertex-triangle contraction  edge-edge contraction  Sufficient to connect the closest points of two objects or parts of an object  Perform no reduction, but increase the connectedness of the model

11. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 11 Generalized Pair Contractions  Vertex-edge contraction  project the vertex onto the corresponding edge  insert an intermediate vertex at the projection point (split the edge and all triangles incident to it)  contract intermediate and contraction vertices

12. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 12  Vertex-triangle contraction  project the vertex onto the corresponding triangle  insert an intermediate vertex at the projection point (split the triangle)  contract intermediate and contraction vertices Generalized Pair Contractions

13. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 13 Generalized Pair Contractions  Edge-edge contraction  find the closest points on the corresponding edges  insert two intermediate vertices at these points (split both edges and all triangles incident to them)  contract two intermediate vertices

14. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 14 Generalized Pair Contractions  Ordering of operations  Quadric error metric [Garland & Heckbert 1997]  Problem: no control of the real geometric error  Problem: error accumulation at flat regions with noise v0v0v0v0 v1v1v1v1 v4v4v4v4 v5v5v5v5 v8v8v8v8 v9v9v9v9 v2v2v2v2 v3v3v3v3 v6v6v6v6 v7v7v7v7 v0v0v0v0 v1v1v1v1 v4v4v4v4 v5v5v5v5 v8v8v8v8 v9v9v9v9 v 10 v 11 Q 10 = Q 2 + Q 3 Q 11 = Q 6 + Q 7 Q 12 = Q 1 + Q 10 Q 13 = Q 5 + Q 11 Q 14 = Q 12 + Q 4 = = Q 1 + Q 2 + Q 3 + Q 4 = Q 1 + Q 2 + Q 3 + Q 4 Q 15 = Q 13 + Q 8 = = Q 5 + Q 6 + Q 7 + Q 8 = Q 5 + Q 6 + Q 7 + Q 8 v0v0v0v0 v9v9v9v9 v 14 v 15 v0v0v0v0 v4v4v4v4 v8v8v8v8 v9v9v9v9 v 12 v 13

15. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 15 Generalized Pair Contractions  Ordering of operations  Non-accumulating error quadrics  Preprocessing phase: calculate initial error quadrics for all vertices  Intermediate vertex created: calculate its quadric from the incident triangles  Contraction operation performed: calculate quadric of a new vertex, recalculate quadrics of all adjacent vertices

16. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 16 Generalized Pair Contractions  Ordering of operations  Error test before the operation:  calculate the local one-sided Hausdorff distance after the operation from the original to the simplified mesh  if this error exceeds the predefined threshold d max, the operation is rejected  Guarantees maximum geometric distance between original and simplified model

17. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 17 Generalized Pair Contractions  Ordering of operations  Approximation of the Hausdorff distance is used to order the possible operations:  - accumulated errors of simplices s 1 and s 2  - local one-sided Hausdorff distance before the operation  - quadric error of the operation  - approximation of the distance between meshes before and after the operation

18. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 18 Generalized Pair Contractions  Handling of sharp features  Feature edges and feature vertices: all incident faces are inside chosen small angle  max  For each feature:  generate average plane P a of all incident faces  generate constraint plane P c running through the feature and perpendicular to P a  add the quadric of P c to the quadrics of appropriate vertices   <  max  PaPaPaPa  PaPaPaPa PcPcPcPc

19. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 19 Generalized Pair Contractions  Handling of sharp features 200 vertices, features not preserved 200 vertices, features preserved Original with detected features

20. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 20 Spatial Search Data Structure  Queries:  for given vertex find closest non- incident simplex  for given edge find closest non- incident edge  need of spatial search data structure  During simplification simplices are eliminated and created  support of fast dynamic updates  Regular grid  average edge length as cell side length

21. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 21 Simplification Algorithm  Preprocessing:  Initialization of the grid  Identification of corresponding pairs  For each vertex the corresponding simplex (vertex, edge or triangle) is found  For some edges the corresponding edge is found  The references to the found simplices are stored  Pairs are inserted into the priority queue according to their errors

22. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 22 Simplification Algorithm  Decimation loop  Pop next operation from the priority queue  Before the operation:  normal test  minimal angle test  error test  collision test  After the operation:  update the grid  update all affected simplex pairs  update the priority queue

23. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 23 Applications and Results  Controlled topology modifying simplification and mesh repair Original model with 4288 vertices Model courtesy of DaimlerChrysler AG

24. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 24 Applications and Results  Controlled topology modifying simplification and mesh repair 500 vertices vertex pair contractions 500 vertices generalized pair contractions

25. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 25 Applications and Results  Controlled topology modifying simplification and mesh repair 125 vertices vertex pair contractions 125 vertices generalized pair contractions

26. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 26 Applications and Results 4467 vertices original 250 vertices gener. pair contractions 250 vertices vertex pair contractions

27. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 27 Applications and Results  Out-of-core simplification  Cutting the model into subparts and simplifying each subpart independently  Gaps are automatically closed when subparts are simplified together Vertex pair contractions Generalized pair contractions

28. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 28 Applications and Results Original 1 087 716 faces OEMM-QEM 18 338 faces Stream decimation 18 486 faces Our method 18 338 faces

29. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 29 Applications and Results  Out-of-core simplification Original 1 087 716 faces OEMM-QEM 18 338 faces error = 0.82% Stream decimation 18 486 faces error = 0.82% Our method 18 338 faces error = 0.26%

30. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 30 Applications and Results  Out-of-core simplification  Error quadrics are not accumulated  better simplification of flat regions with noise OEMM-QEM 18 338 faces error = 0.82% Stream decimation 18 486 faces error = 0.82% Our method 18 338 faces error = 0.26%

31. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 31 Conclusion  A strategy to generate high quality approximations for models with arbitrary topology  Generalized pair contraction operations introduced  Allow to remove gaps and holes and integrate the automatic connection of close surface parts  Guaranteed maximum geometric distance between original and simplified model  Simplification controlled by a geometric error

32. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 32 Future Work (already done)  Intersection free simplification  Out-of-core simplification  View-dependent out-of-core visualization

33. P. Borodin, S. Gumhold, M. Guthe, R. Klein - High-Quality Simplification with Generalized Pair Contractions University of Bonn  Computer Graphics Group 33 Thank you